On Mar 17, 10:16 am, "Dave L. Renfro" <renfr...@cmich.edu> wrote:> Kerry Soileau wrote:> > Let P be the collection of all partitions of the natural numbers.> > A partition of a set X is a pairwise disjoint collection of> > sets whose union is X. I have a proof that the cardinality of P> > is the cardinality of the real numbers. Is this well-known?> > If so, would someone provide a reference?>> This seems to me to be around the average difficulty level of> a problem in an undergraduate set theory or real analysis text,> so it's not something anyone would bother citing a reference> to if they wanted to use the result somewhere (and off-hand,> I don't know of a specific reference, and all my books are at> home and I'm elsewhere).

Exercise 6 on p. 76 of W. Sierpinski, _Cardinal and Ordinal Numbers_,Second Edition Revised, Warszawa, 1965: "Prove that if X is the set ofall natural numbers, then the set Z of all partitions of the set N iseffectively of the power of the continuum." Here "effectively of thepower of the continuum" means that the equivalence is established byan effectively definable bijection. The author provides a detailedsolution on the same page.

> To show that there are at least continuum many possible partitions> of N, let E be the set of positive even integers. Then, for each> subset B of E, the set {B, N - B} is a partition of N. Moreover,> if B, B' are distinct subsets of E, then {B, N - B} is not equal> to {B', N - B'}. Thus, there are at least continuum many partitions,> since there are continuum many subsets of E.

Alternatively, you can associate with each real number a partition ofthe set Q of rational numbers into two classes (Dedekind cuts), andthen use a bijection between Q and N.